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Theorem toponcom 22774
Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 22760 . . . 4 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐽 = βˆͺ 𝐾)
21eqcomd 2730 . . 3 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐾 = βˆͺ 𝐽)
32anim2i 616 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
4 istopon 22758 . 2 (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾) ↔ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
53, 4sylibr 233 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4900  β€˜cfv 6534  Topctop 22739  TopOnctopon 22756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-topon 22757
This theorem is referenced by:  toponcomb  22775  kgencn3  23406
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